In this study, we present unified and closed-form analytical expressions of wave functions, Wigner distribution function, and characteristic function associated with a class of generalized Laguerre polynomials. These formulations cover three known fundamental quantum potentials, namely, the Morse potential, the three dimensional harmonic oscillator and the three dimensional Coulomb potential. By performing the change of variable in the phase space, we derive identical expressions for higher-order moments due to a canonical transformation linking the two phase space. We introduce the quantity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\langle g^m(x)\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>g</mi> <mi>m</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, related to generalized Laguerre potentials, to evaluate the expectation values of potential functions. Additionally, we compute the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\eta _{\xi ,\alpha _\xi }(n,L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>η</mi> <mrow> <mi>ξ</mi> <mo>,</mo> <msub> <mi>α</mi> <mi>ξ</mi> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> expressed via derivatives of hypergeometrical functions evaluated at zero, which allowed us to determine the normalization constant of the wave function associated with physical potentials. The obtained solutions reproduce the well-established results associated with the Morse, 3D-harmonic oscillator, pseudo-harmonic, 3D-Coulomb, and Hartmann potentials, thereby confirming the consistency and robustness of the present framework.

Class of generalized Laguerre potentials in the phase space representation

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发表日期：

2026年1月9日

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Othmane Cherroud,
Sid-Ahmed Yahiaoui

摘要

In this study, we present unified and closed-form analytical expressions of wave functions, Wigner distribution function, and characteristic function associated with a class of generalized Laguerre polynomials. These formulations cover three known fundamental quantum potentials, namely, the Morse potential, the three dimensional harmonic oscillator and the three dimensional Coulomb potential. By performing the change of variable in the phase space, we derive identical expressions for higher-order moments due to a canonical transformation linking the two phase space. We introduce the quantity $\langle g^m(x)\rangle $ $⟨ g^{m} (x) ⟩$ , related to generalized Laguerre potentials, to evaluate the expectation values of potential functions. Additionally, we compute the function $\eta _{\xi ,\alpha _\xi }(n,L)$ $η_{ξ, α_{ξ}} (n, L)$ expressed via derivatives of hypergeometrical functions evaluated at zero, which allowed us to determine the normalization constant of the wave function associated with physical potentials. The obtained solutions reproduce the well-established results associated with the Morse, 3D-harmonic oscillator, pseudo-harmonic, 3D-Coulomb, and Hartmann potentials, thereby confirming the consistency and robustness of the present framework.